Methods and apparatus for reducing error in interferometric imaging measurements

ABSTRACT

Described is a fringe generator for an interferometric measurement system having improved fringe stability and reproducibility. The fringe generator includes a light source at a characteristic wavelength and a diffractive element to generate a pair of diffracted beams from light received from the light source. The fringe generator also includes a lens to receive the pair of diffracted beams and to image the plane of the diffractive element onto an object to be measured. The generated fringe pattern is substantially independent to a change in the position of the light source relative to the lens and a change in the characteristic wavelength of the light source. A broadband light source can be used and the resulting broadband fringe pattern is substantially independent to a change in the position of the light source relative to the lens and to a change in the spectral distribution of the broadband light source.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application is a divisional of U.S. patent application Ser. No.10/871,878 filed on Jun. 18, 2004, which claimed the benefit andpriority to U.S. Provisional Patent Application Ser. No. 60/479,534filed on Jun. 18, 2003. The entireties of the above-referencedapplications are incorporated by reference herein.

FIELD OF THE INVENTION

The present invention relates generally to the field of imagingtechnology and, more specifically, to three-dimensional imaging methodsand devices.

BACKGROUND OF THE INVENTION

Measuring the characteristics of an object and generating a threedimensional representation of the object from acquired sensor data arecentral objectives in the field of metrology. The continuing developmentof various techniques to achieve these objectives is often grounded inusing interferometric principles to obtain precise measurement data. Inparallel with the development of fundamental imaging and metrologytechnologies for science and industry, refining and improving uponexisting approaches represents a valuable and necessary contribution tomodern day research efforts. Thus, given the wide use of interferometricapproaches, it follows that a need exists for methods and devices thatreduce errors in present and future interferometric based imaging andmetrology applications.

SUMMARY OF THE INVENTION

The present invention provides a method and apparatus for reducing errorin interferometric fringe stability and reproducibility in aninterference fringe generator. In one aspect, the method for reducingerror in interferometric fringe stability and reproductivity includesproviding a light source, positioning a grating to receive light fromthe light source and positioning a projection lens having a focal lengthF to receive light from the grating. The projection lens projects thereceived light upon an object of interest positioned substantially at adistance d₁ from the lens. Typically the lens is positionedsubstantially at a distance d₂ from the grating. The values of d₁, d₂,and F are related by d₂ being approximately equal to d₁F/(d₁−F).

Generally, the terms fringe projector and fringe generator are usedinterchangeably throughout the application. Such devices alone or incombination with other components cause the formation of interferencefringes in space. Such fringes are typically generated or projected uponthe surface of an object of interest. These devices differ from standardimage projectors that do not utilize interference principles.

In another aspect, the invention relates to an optical system forreducing error in interferometric fringe stability and reproducibilityin an interference fringe generator. The system typically includes alight source, a grating positioned to receive light from the lightsource and a lens for projecting fringes having a focal length F. Thelens is positioned to receive light from the grating and to project thereceived light upon an object of interest positioned substantially at adistance d₁ from the lens. Typically, the lens is positionedsubstantially at a distance d₂ from the grating. The values of d₁, d₂,and F are related by d₂ being approximately equal to d₁F/(d₁−F). Invarious embodiments the light source is a coherent light source such asa laser. In other embodiments an acousto-optical modulator or otherdiffractive element is used in lieu of a grating.

In another aspect, the invention relates to a method for reducing errorsin positional information obtained from a translatable grating in aninterference fringe generator. The method includes positioning atranslatable grating substantially defining a first plane within afringe projector. Positioning an encoder scale defining a second planesubstantially orthogonal to the first plane directly over the grating isanother step in the method. The method also includes measuring therelative motion of the grating with respect to the encoder scale. Invarious embodiments the grating and encoder scale are manufactured onthe same substrate. In various embodiments the grating and encoder scaleare made from the same material. In one embodiment of the invention theencoder read-head is aligned with the optical axis of the system'sprojection lens.

In yet another aspect, the invention relates to a method for reducingerrors in positional information obtained from a translatable grating inan interference fringe generator. The method includes positioning atranslatable grating substantially defining a first plane within afringe projector. The method also includes positioning two encoderscales substantially parallel to the first plane at positions above andbelow the grating. Measuring the relative motion of the grating withrespect to the encoder scales and using differential information fromthe relative motion of the grating with respect to the encoder scales toreduce pitch based errors are also steps in the disclosed method.

Other aspects and advantages of the present invention will becomeapparent from the following drawings, detailed description, and claims,all of which illustrate the principles of the invention, by way ofexample only.

BRIEF DESCRIPTION OF THE DRAWINGS

The foregoing and other objects, features, and advantages of theinvention described above will be more fully understood from thefollowing description of various embodiments, when read together withthe accompanying drawings. In the drawings, like reference charactersgenerally refer to the same parts throughout the different views. Thedrawings are not necessarily to scale, and emphasis instead is generallyplaced upon illustrating the principles of the invention.

FIG. 1 is a schematic representation of an embodiment of a portion of aninterference fringe generating system in accordance with the teachingsof the invention;

FIG. 2 illustrates an embodiment of an interference-fringe projectorsuitable for use with various aspects of the invention;

FIG. 3 illustrates the characteristics of a diffraction grating suitablefor use with various aspects of the invention; and

FIGS. 4A and 4B are schematic representations of differing measurementsystem configurations that illustrate various techniques and approachesfor reducing measurement error in accordance with the teachings of theinvention.

DESCRIPTION OF THE PREFERRED EMBODIMENTS

In various metrology applications and imaging systems, interferenceprinciples form the underlying basis upon which the application orsystem operates. In one exemplary imaging apparatus two point sources oflight are employed to produce interference fringes. Typically coherentlight is used, such as for example a laser. These two point sources mayderive from an original source that was previously split or otherwisemodified by an arrangement of one or more optical elements, such as adiffractive element. In systems of this type, light from the two pointsources is directed and manipulated such that it expands and overlaps.As a result of this interaction, controlled sinusoidal fringes, alsoknown as accordion fringes, are produced throughout space.

As the generation of these fringes is controlled, various parametersregarding the fringes as well as the apparatus and methods whichproduced them are known. Thus, when the fringes impinge on an object,surface profile and dimensional information about that object may becalculated after the fact. In one embodiment this is achieved by imagingthe fringes, typically with a detector such as a conventional camera orCCD array, optionally changing the fringes according to a prescribedprocess, and finally calculating point cloud information about theobject. From this it is clear that the positional stability andrepeatability of the fringes is important when making high qualitymeasurements.

One factor in the design of interferometric imaging and metrologydevices or processes that often results in positional stability fringeerror is movement in the fringe producing light source. In part, theinvention is directed to both apparatus and methods for mitigating theeffects of light source angular movement that contribute to undesirablefringe pattern motion.

FIG. 1 shows a schematic representation of the fringe generating portion100 of an imagining device. At a general level, in the embodiment shown,the fringe generating portion 100 includes a light source 105, adiffractive element, shown as grating 110, and a projection lens 115.The projection lens 115 has a focal length F and the grating has aperiod T. Although a grating 110 is shown in this embodiment, otherdiffractive elements can be used instead, such as an acousto-opticalmodulator. The optical axis 117 of the projection lens 115 is designatedby the dotted line shown. The incident angle of the system's lightsource 105, shown here as a laser, with respect to the optical axis 117of the projection lens is θ. As shown in the figure, the longitudinalseparation between the grating 110 and projection lens 115 is d₂ and thedistance between the projection lens and the region of measurement 120for the object of interest is d₁.

A straightforward diffraction analysis reveals that the sinusoidalfringe irradiance profile at the object location is given by I(x) where:

${I(x)} = {\cos^{2}\left\lbrack {\frac{2\pi}{T}\left\{ {{\left( \frac{F}{d_{1} - F} \right)x} - {\left( {d_{2} - \frac{d_{1}F}{d_{1} - F}} \right)\theta}} \right\}} \right\rbrack}$It is noteworthy that as θ changes, the phase of the sinusoidal fringepattern changes. In other words, the position of the sinusoidal fringepattern is a function of the incident angle of the laser beam, θ, in thesystem. This angular dependence of the fringe pattern can be eliminatedby proper choice of the distances d₁ and d₂ for a given projection lens115 focal length F. Thus, if the longitudinal distance between theprojection lens 115 and the grating 110 is properly set, then anyangular movement of the laser beam 105 that impinges on the grating willnot result in fringe pattern motion. Eliminating errors introduced bylight source 105 motion will in turn improve metrology and imaging dataquality in various types of systems and devices using fringe generationbased techniques.

After making this discovery, it becomes desirable to evaluate the θdependent term in the I(x) function discussed above. Further analysisreveals that the θ dependence will be eliminated if d₂ is set such that:

$d_{2} = \frac{d_{1}F}{d_{1} - F}$Rearranging this equation, results in the following lens formularelationship:

${\frac{1}{d_{1}} + \frac{1}{d_{2}}} = \frac{1}{F}$Thus, this algebraic manipulation reveals that the θ dependence of thesystem will be eliminated if the system is set-up such that theprojection lens 115 images the plane of the grating 110 out to the planeof the object being measured. Although the aspects of the invention arediscussed in relation to specific illustrative embodiments, the generalfeatures of the invention relating to projection lens and gratingpositioning apply to all imaging systems and metrology techniques whereinterferometric fringes are generated upon the surface of an object ofinterest.

Referring now to FIG. 2, therein is shown an embodiment of a broadbandor white-light interference fringe projector. Light source 250 generatesa substantially collimated beam of radiation 252 that is directed totranslatable diffraction grating 254 at substantially normal incidencebefore passing to projection lens 266. The aspects of the inventiondiscussed above in relation to FIG. 1 are directly applicable to thesource 250, grating 254, lens 266 arrangement substantially at positionsd₁ and d₂ as shown in this embodiment.

It should be noted that the fringe-generation scheme depicted in FIG. 2can also produce fringes using narrow-band or laser illumination whileincorporating the features of the invention discussed in relation toFIG. 1. One advantage of using diffraction grating 254 followed by lens266 for narrow-band illumination is that fringe period d is insensitiveto wavelength so that frequency drifts of the source do notsubstantially degrade measurements. For example, although laser diodesare relatively inexpensive and readily available sources, they have atemperature-dependent operating wavelength. However, since thistechnique is insensitive to temperature-dependent wavelength shifts,laser diodes can be used without measurement degradation.

Again referring to FIG. 2, diffraction grating 254 is shown with gratingperiod D. Input beam 252 is represented by three constituent wavelengthcomponents λ₁, λ₂, and λ₃ for illustration. Beam 252, in actuality, canhave arbitrary spectral composition. Diffraction grating 254 splits beam252 into multiple diffracted beams whose diffraction orders can berepresented by the integer m. For illustration purposes, only rays alongthe perimeter of beam 252 are shown. These diffracted beams propagate atangles θ_(m) with respect to the optical axis 258 according to thegrating equation for normal incidence, which is given by:

${\sin\;\theta_{m}} = \frac{m}{\lambda\; D}$

In one embodiment, diffraction grating 254 is designed to maximize andequalize the diffracted efficiency for the diffraction order m=+1diffracted beam 260 and the diffraction order m=−1 diffracted beam 262.In other embodiments, diffraction grating 254 is designed to maximizeand equalize the diffracted efficiency for any set of positive andnegative beams of equal order |m|, and to minimize the energy diffractedinto all other orders. Any residual undiffracted (m=0) beam 264, willpass undeviated through diffraction grating 254 and is focused byprojection lens 266 onto focal spot 268.

The spectral components λ₁, λ₂, and λ₃ of focal spot 268 substantiallyoverlap. Focal spot 268, in one embodiment, may be substantially blockedby the central obstruction 270 of optional double slit 272. Thedifferent spectral components λ₁, λ₂, and λ₃ of diffracted beams 260 and262 are focused by lens 266 onto spectral regions 274 and 276. Thedistance α(λ) between the focal spot within spectral region 274 and thefocal spot within spectral region 276 corresponding to a givenwavelength λ is substantially proportional to the wavelength λ. Aperturestop 278 of lens 266, in one embodiment, can be used to block undesiredhigher-order diffracted beams. In other embodiments the aperture stop isnot used. Any undesired residual diffracted orders that pass throughlens 266 can be blocked, in another embodiment, by the opaque regions ofoptional double slit 272. Radiation from the two spectral regions 274and 276 expands and overlaps as it propagates and forms aninterference-fringe pattern 280. Fringe pattern 280 has representativefringe period d at representative distance R from double slit 272.

In one embodiment, diffraction grating 254 is a thin phase gratinghaving a square-wave phase profile whose relative phase delay alternatesbetween 0° and 180° for a representative wavelength, λ₂, with a 50% dutycycle. Although in various embodiments any suitable grating can be used.Grating 254 is relatively efficient, diffracting approximately 40.5% ofthe available energy into each of the m=−1 and m=+1 diffracted orders,and nominally 0% into the m=0 and other even diffracted orders. Therelative phase delay of grating 254 is a function of wavelength, causingthe energy in the undiffracted beam 264 at order m=0 to increase forwavelengths that differ from the representative wavelength λ₂.

Phase shifting the resulting broadband (or narrow-brand)interference-fringe pattern 280 is achieved by simply translatingdiffraction grating 254 in the direction 282 shown in FIG. 2.White-light or broadband phase shifting is realized because atranslation of diffraction grating 254 by a given fraction of thegrating period D shifts each spectral component of fringe pattern 280 bytwice the same fraction of the fringe period d. For example, atranslation of grating 252 by D/4, or one-quarter cycle, also shifts theinterference-fringe pattern 280 by one-half cycle.

Accordion motion (or variation of fringe size) of interference-fringepattern 280 can be achieved in a number of ways. In one embodiment, forsmall diffracted angles θ_(m), doubling the period D of grating 254halves the magnitude of θ_(m) of beams 260 and 262 which in turn doublesthe period d of fringe pattern 280. In another embodiment, decreasingthe focal length f of lens 266 can increase the period d of fringepattern 280. However, no matter how changes in the fringe pattern areactuated, the placement of the lens, grating, and object of interest astaught herein contributes to producing fringe stability andreproducibility.

Although methods for reducing the impact of light source movement onfringe motion are discussed above, another aspect of the inventionrelates to reducing the error when fringe motion is caused by design ina controlled fashion as part of the process of imaging an object. One ofthe operational features in various fringe generating imaging systemsinvolves precisely translating a grating over specified distances. Insome embodiments these translation distances are approximately 50 mm.Generally, the only limitations on translation distance are a functionof a given imaging system set up and the size of the diffractiongrating. In various embodiments, the accuracy with which the lateralposition of the grating needs to be known is approximately 1/10 of amicrometer. In general various aspects of the invention relate toaccurately knowing the position of the grating because the position ofthe projected fringes is directly correlated with the position of thegrating. This is discussed above in relation to translatable grating 254in FIG. 2.

FIG. 3 schematically illustrates a grating 110′ and an encoder scale 300according to an embodiment of the invention. The discussion of grating110′ shown in FIG. 2 also applies to the grating 110 in FIG. 1 and thegrating 254 in FIG. 2. Generally, the discussion of FIGS. 3, 4A and 4Bapplies to those embodiments wherein a grating is translated to changethe characteristics of a fringe pattern as part of an imaging ormetrology system.

In various interferometric measuring systems, a commercial linearencoder can be used to determine the position of the grating (110, 110′,254). Other types of measurement devices employing fixed scales anddevices for reading those scales can also be used in other embodiments.The linear encoder operates by monitoring the position of an encoderscale 300 with an encoder read-head (not shown). One of the largestsources of error that is typically encountered when using a linearencoder is Abbé error. Abbé error occurs when the spatial position ofinterest is displaced from the true position recorded by the measuringsystem. This in turn results in the introduction of errors in any systemusing the data generated by the measuring system.

Still referring to FIG. 3, in some interferometric measuring systemsembodiments, Abbé error is encountered when the object for whichpositional information is sought (in our case the grating 110′) is notat the same location as the position measuring device. In theillustrative embodiment shown in FIG. 3, this measuring device is alinear encoder read-head. As the object (grating 110′) is translated,any relative angular changes between the measuring device and thegrating 110′ will result in Abbé error.

Again referring to FIG. 3, the position 305 on the encoder scale 300represents the location where the encoder read-head is making a positionmeasurement. The position 310 on the grating 110′ represents thelocation that must be tracked and measured to obtain overall gratingpositional information. Typically, in various embodiments the grating110′ and encoder scale 300 are rigidly coupled₁ and are translated by amotorized stage (not shown) along the stage travel axis shown in theFIG. 3. As the stage is translated, the grating 110′ and encoder scale305 both incur small angular rotations due to imperfections in thestage. Two of these angular rotations, pitch and yaw, will introduceAbbé error.

The easiest way to visualize this is to assume that the axes of thepitch and yaw rotations intersect at the encoder read-head location.(This is not a necessary condition for Abbé error to occur. It is onlyassumed for visualization purposes.) In this case, as the stage pitchesand yaws, the encoder read-head value does not deviate. However, thegrating position 310 will change due to the pitch and yaw. The amountthat the grating position changes (see FIG. 3) is approximatelyd_(pitch)θ_(pitch) in the pitch direction and d_(yaw)θ_(yaw) in the yawdirection. Thus in order to reduce the grating position changes, thegrating—encoder scale distances, d_(pitch) and d_(yaw), must beminimized or reduced as much as possible.

FIGS. 4A and 4B illustrate some techniques according to the inventionfor reducing Abbé error. Configuration 1 shown in FIG. 4A is applicableto various gratings such as the one shown in FIG. 3. The encoder scaleis mounted perpendicular to the grating and directly over it. Thisconfiguration typically eliminates the yaw component of the Abbé error,and minimizes the pitch component. Note that the pitch component is notcompletely eliminated, because the encoder scale cannot occlude thegrating if the system is to operate properly. In one embodiment of theinvention the encoder read-head is aligned with the optical axis of thesystem's projection lens. In some embodiments the grating 110′ andencoder scale 300 are fabricated from the same materials to facilitate areduction of temperature-based errors. These two system aspectseliminate imaging system errors that would otherwise be present due totemperature changes in the various imaging and metrology systems.

The implementation of Configuration 1 in FIG. 4A does not completelyeliminate the Abbé error due to stage pitch. Another proposedimplementation (See Configuration 2 in FIG. 4B), may eliminate orsubstantially reduce the Abbé error due to pitch. This configurationuses two linear encoder scales 300A, 300B (and one or more measuringdevices, typically two encoder read-heads). The measurement scales arepositioned with one above and one below the grating 110′ as shown.Knowing the relative positions of the two read-heads with respect to thegrating 110′, it is possible to take the two readings from the twoencoders 300A, 300B and calculate the pitch error at the grating center.This type of differential measuring enables differences between the twoencoder scales readings to facilitate a determination of pitch error. Insome embodiments a processor, such as a computer processor or logiccircuit, carries out the pitch error and yaw error calculations. As withConfiguration 1, the encoder scales 300A, 300B lie in the same plane asthe grating 110′, so that the yaw error is approximately zero. As onevariation of Configuration 2, the encoder scales and gratings arefabricated on a single substrate in various embodiments. This providesimproved alignment and enhanced thermal stability. Further errorreduction can be achieved by combining the general aspects of theinvention relating to encoder scale and optical element positioning inone device or measurement system.

Having described and shown the preferred embodiments of the invention,it will now become apparent to one of skill in the art that otherembodiments incorporating the concepts may be used and that manyvariations are possible which will still be within the scope and spiritof the claimed invention. It is felt, therefore, that these embodimentsshould not be limited to disclosed embodiments but rather should belimited only by the spirit and scope of the following claims.

1. A fringe generator for an interferometric measurement system havingimproved fringe stability and reproducibility comprising: a light sourcehaving a characteristic wavelength; a diffractive element disposed toreceive light from the light source and to generate a pair of diffractedbeams; and a lens having an optical axis, the lens being positioned toreceive the pair of diffracted beams from the diffractive element and toimage a plane of the diffractive element onto an object to be measured,wherein a fringe pattern generated on the object to be measured issubstantially independent to a change in position of the light sourcerelative to the optical axis and a change in the characteristicwavelength of the light source.
 2. The fringe generator of claim 1wherein the pair of diffracted beams have a complementary diffractionorder.
 3. The fringe generator of claim 1 wherein the diffractiveelement comprises a diffraction grating.
 4. The fringe generator ofclaim 1 wherein the diffractive element comprises an acousto-opticalmodulator.
 5. The fringe generator of claim 1 further comprising adouble aperture disposed between the lens and the object to be measuredwherein the pair of diffracted beams propagates through the doubleaperture and other diffracted beams generated by the diffractive elementare blocked.
 6. The fringe generator of claim 1 further comprising acentral obscuration disposed between the diffractive element and theobject to be measured wherein an undiffracted beam propagating from thediffractive element is blocked.
 7. The fringe generator of claim 1wherein the light source comprises a laser.
 8. The fringe generator ofclaim 1 further comprising a translation stage coupled to thediffractive element and adapted for translation of the diffractiveelement to generate a phase shift in an interference fringe patterngenerated by the fringe generator.
 9. A broadband fringe generator foran interferometric measurement system having improved fringe stabilityand reproducibility comprising: a broadband light source having aspectral distribution; a diffractive element disposed to receive lightfrom the broadband light source and to generate a pair of spectrallydistributed diffracted beams; and a lens having an optical axis, thelens being positioned to receive the pair of spectrally distributeddiffracted beams from the diffractive element and to image a plane ofthe diffractive element onto an object to be measured, wherein abroadband fringe pattern generated on the object to be measured issubstantially independent to a change in position of the light sourcerelative to the optical axis and a change in the spectral distributionof the broadband light source.
 10. The broadband fringe generator ofclaim 9 wherein the pair of spectrally distributed diffracted beams hasa complementary diffraction order.
 11. The broadband fringe generator ofclaim 9 wherein the diffractive element comprises a diffraction grating.12. The broadband fringe generator of claim 9 wherein the diffractiveelement comprises an acousto-optical modulator.
 13. The broadband fringegenerator of claim 9 further comprising a double aperture disposedbetween the lens and the object to be measured wherein the pair ofdiffracted beams propagates through the double aperture and otherdiffracted beams generated by the diffractive element are blocked. 14.The broadband fringe generator of claim 9 further comprising a centralobscuration disposed between the diffractive element and the object tobe measured wherein an undiffracted beam propagating from thediffractive element is blocked.
 15. The broadband fringe generator ofclaim 9 further comprising a translation stage coupled to thediffractive element and adapted for translation of the diffractiveelement to generate a phase shift in an interference fringe patterngenerated by the broadband fringe generator.